In this note the conditions on an invariance principle for triangular arrays of random variables contained in an earlier paper are weakened. Random norming by functions which are not stopping times is permitted, the $L^2$-boundedness conditions on the maximum of the summands relaxed, and joint convergence with an arbitrary sequence of random elements of some other metric space proved.
"An Extended Martingale Invariance Principle." Ann. Probab. 6 (1) 144 - 150, February, 1978. https://doi.org/10.1214/aop/1176995619